The 24-year-old forward has been a regular source of goals in the Scottish Football League over recent years and will now get the chance to push for a dream move to England.
A number of SPL sides were believed to be keen on taking a look at Doris, who recently caught the eye during a cup meeting with Celtic, but Birmingham have edged to the front of the queue.
Doris is to spend a week training with the Championship club and Arbroath boss Paul Sheerin is convinced his main source of goals can be a success at St Andrew's.
He said in the Daily Record: "Birmingham watched him a few times and saw him against Celtic. Now they want a closer look.
"He's been outstanding for the past few years and I'm surprised more Scottish clubs haven't had a nibble at him.
"He's good enough to step up and I know he won't let himself down when he goes to Birmingham."